Posted: December 18th, 2015

Two Dimensional Hyperbolic Equation Expand the explicit method for hyperbolic equation to two dimensions and solve: Two Dimensional Wave Vibration Over Square Membrane

mathematics
Assignment 7
Elliptic PDE

Problem #1: Alternating Direction Implicit (ADI) Method
Research and write a brief explanation of the ADI Method and solve:
Two Dimensional LaPlace Equation: Electric Potential Over a Flat Plate with Point
Charge
!!!
∇2
u(x, y) = f (x, y) for -1 ≤ x ≤ 1, -1 ≤ y ≤ 1
boundary conditions: u(x,y) = 0 for all boundaries
f (0.5,0.5) = −1
f (−0.5,−0.5) = 1
elsewhere : f (x, y) = 0
Two Dimensional Temperature Diffusion:
!!!
10−4 ∂2
u(x, y,t)
∂x2 +
∂2
u(x, y,t)
∂y2





⎟ = ∂u(x, y,t)
∂t
for 0 ≤ x ≤ 4, 0 ≤ y ≤ 4 0 ≤ t ≤ 5000
u(x, y,0) = 0
u(x, y,t) = ey cos x − ex cos y for x = 0, x = 4, y = 0, y = 4
! ! Present results for t= 5000
Problem 2: Crank-Nicolson Problem
Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method.
Assignment 8
ENGR516 Assignment #5: Hyperbolic PDEs
Problem 1: Two Dimensional Hyperbolic Equation
Expand the explicit method for hyperbolic equation to two dimensions and solve:
Two Dimensional Wave Vibration Over Square Membrane
!!!
0.25 ∂2
u(x, y,t)
∂x
2 +
∂2
u(x, y,t)
∂y
2





⎟ = ∂2
u(x, y,t)
∂t
2
for 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 and 0 ≤ t ≤ 2
u(0, y,t) = 0, u(2, y,t) = 0
u(x,0,t) = 0, u(x,2,t) = 0
u(x, y,0) = 0.1sin(π x)sin(π y / 2), ∂u(x, y,0)
∂t = 0
! ! Present Results for t = 0.1 and t = 1.8
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